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The aim of this work is to provide an algebraic-geometric method to construct generalized local symbols on curves as morphisms of group schemes. From a closed point of a complete, irreducible, and nonsingular curve C over a perfect field k as the only data, using theta groups over Picard schemes of curves, we offer a geometric construction that allows us to define generalizations of the tame symbol and the Hilbert norm residue symbol. [ABSTRACT FROM AUTHOR]